theoretical physics part ii: lattice gauge theorymeurice/doe/ymeuricedoevisit2012.pdflattice models...

Theoretical Physics Part II: Lattice Gauge Theory

Yannick Meurice

The University of Iowa

[email protected]

with Don Sinclair (ANL/U.of Iowa),

Alan Denbleyker, Yuzhi “Louis” Liu, Judah Unmuth-Yockey, and Haiyuan Zou

Iowa City, October 22, 2012

Yannick Meurice (U. of Iowa) Theoretical Physics (Meurice) Iowa City, October 22, 2012 1 / 87

Content of the presentation

OverviewScientific goalsPeopleActivitiesFacilitiesRecent progress

1 Toward a Fisher’s zero approach of the conformal window (EnergyFrontier)

2 B-physics beyond the standard model (Intensity Frontier)3 New methods in lattice field theory

Models of a Strongly-Coupled Higgs Sector (by Don Sinclair)


Overview

Main interest: Models of strong interactions primarily on the latticeApplications: QCD and extensions beyond the standard modelMethods: MC simulations, improved perturbation theory andrenormalization group techniquesSenior Personnel: Don Sinclair (1/4 time at U. of Iowa, DOEfunded)Graduate Students: 4 at U. of Iowa now (one of them at Fermilabwith URA for AY 12-13), one recently graduated (now postdoc atU. Illinois Urbana Champaign)Computational facilities: Clusters here and at Fermilab, NERSCNew computational possibilities explored: optical latticerealizations of lattice gauge theory (NOT DOE funded)


Success of the Standard Model (SM)

Tree level gauge boson masses: M2W = πα√

2GF sin2θW= M2

Z cos2θW

Low-energy data can be used to determine the input parameters(α, GF and sinθW )1982: W and Z discovered with consistent massesPrecision measurements (LEP, ...) + radiative corrections implythat mt ≈ 160GeV/c2 (around 1990)Evidence for top quark (around 1994); 172 GeV/c2 (2010)All the above + radiative corrections: 40 < MH < 160GeV/c2

(typical values found in PDG since 1999)2011: Tevatron and LHC exclude most of the above region2012: LHC discovery with MH ' 125Gev/c2 ?No consistent and sustained hints for beyond the SM physics


2012: consistent hints for MH ' 125Gev/c2 at LHC

Figure: Most recent CMS σ/σSMH graphs (July 2012)


Important problems in particle physics

Masses and mixing angles for neutrinosAccurate calculations of weak matrix elements; essential toobserve hypothetical discrepancies with the SMError estimates for applications of perturbative QCDPhase diagram of QCD at finite temperature and densityModels for a light composite Higgs coming from hypothetical newgauge interactions at a multi-TeV scale; the difference in scalesuggests an approximate conformal symmetryWhy 3 generations?Gravity at short distance?

Note: around 1985, the HEP community put more emphasis on logicalproblems than on dynamical or computational problems. In 2012,dynamical and computational problems stand out.


The role of new theoretical methods in the next decade

Recent LHC results indicate that better methods to deal with stronginteractions will become crucial to test the standard model or offereconomical alternative to the standard Higgs mechanism. The onlynonperturbative formulation known to define QCD or QCD-like theoriesis the lattice regularization.

Monte Carlo simulations provide robust results for lattice gaugemodels. The field has been driven by fast progress in CPU and GPU,but the lattices remain small and the lattice spacing large, requiringdifficult extrapolations. Ultimately, we need to find more analyticalmethods to understand the continuum limit and the infinite volume limitof these models especially in near conformal situations.


Lattice models


Lattice Models Considered I: O(N) nonlinear sigmamodels

The lattice sites are denoted x and the scalar fields ~φx areN-dimensional unit vectors. The partition function reads:

Z = C∫ ∏

x

dNφxδ(~φx.~φx − 1)e−βE [{φ}] ,

withE [{φ}] = −

∑x,e

(~φx.~φx+e − 1) ,

with e running over the D positively oriented unit lattice vectors andβ ≡ (1/g2

0)


Lattice Models Considered II: U(1) and SU(N) LatticeGauge Theory

The unitary matrices Ulink are associated with the links (or bonds) of acubic lattice.

Z =∏links

∫dUlink e−βS ,

with the Wilson action

S =∑

p

(1− (1/N)ReTr(Up)) .

and β ≡ 2N/g2. Up denotes the ordered products of 4 Ulink along anelementary square (“plaquette"). We typically use periodic boundaryconditions.


Methods

Monte Carlo simulations (facilities are discussed below)We have made analytical progress in two directions:

Improving Feynman diagrams methods (control of the large fieldcontributions in the path integral)Improving Renormalization Group methods (removing the “walls" inblocking procedures)

Developing new methods requires to go up on a “ladder" of latticemodels (Integrals→ Quantum mechanics→ 2D Ising model→.....→ 4D Gauge theories with fermions (see below)).


The Renormalization Group(RG) method


The lattice theory ladder


The conformal window

The possibility of having a strongly coupled and composite Higgssector has motivated searches for nontrivial infrared fixed points inasymptotically free gauge theories.The location of the conformal windows, the region in parameterspace where a nontrivial infra-red fixed point exists, for severalfamilies of models has been the subject of many recentinvestigations. When approached on the low Nf side this is astrongly interacting problem that can only be treated with LatticeGauge Theory.A situation of particular phenomenological interest is when the βfunction for a new hypothetical gauge coupling approaches zerofrom below and starts “walking".


Complex RG flows

The near conformal situation of a walking coupling constant canbe obtained in some examples by varying a parameter in such away that two RG fixed points coalesce and disappear in thecomplex plane.This motivated us to study extensions of the RenormalizationGroup (RG) flows in the complex coupling plane.In all examples considered, we found that the Fisher’s zeros act as“gates” for the RG flows ending at the strongly coupled fixed point.The Fisher’s zeros are the zeros of the partition function in thecomplex inverse coupling or inverse temperature plane (later, weuse the “β-plane" terminology).This is a complex extension of the general picture of confinementproposed by Tomboulis.


Senior Personnel: Don Sinclair

Pioneer in finite temperature QCD, composite Higgs models andnumerical methods to calculate fermion determinantsSpecial Time Appointment (< 50 percent) at Argonne25 percent appointment at U. of Iowa, funded by the DOEsinceJuly 2012Recent work: SU(3) with 2 and 3 sextets (next talk)Shares codes and computational facilities with usPlayed an essential in our recent calculations of Fisher zeros forSU(3) with 4 and 12 quarks flavors


Daping Du (former graduate student)

Daping Du came in fall 2005 and earned his Ph. D. degree in June2011. Attended the Seattle Lattice Gauge Theory summer school in2007. He worked with the Fermilab Lattice group with a URA fellowshipfrom January to August 2011 and calculated the fragmentationfractions for the B meson. He is now a postdoc at the University ofIllinois in Urbana. He worked on the fits of plaquette distribution,saddle point estimates of the Fisher zeros and interpolations for thedensity of states in U(1) and SU(2) gauge theories. He developednew algorithms for histogram reweighting and search for zeros.


Alan Denbleyker (graduate student)

Alan Denbleyker came in fall 2006. Attended the Seattle Lattice GaugeTheory summer school in 2007. He works on MC simulations in SU(2)gauge theories with and without adjoint terms and works on histogramreweighting and finite size scaling to compare finite temperature andbulk transitions. He is the system manager for our cluster andrepository. Has been supported as a T.A. during the academic yearand as a R.A. during summer. He has passed the qualifying exam andwill take the comprehensive exam soon. This year he is R. A. insummer and fall. Graduation expected in May 2013.

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uette

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uette


Yuzhi “Louis" Liu (graduate student)

Yuzhi “Louis" Liu came in fall 2006. He has passed the qualifying andcomprehensive exams and the T.A. certification. Attended the LesHouches Lattice Gauge Theory summer school in 2009 andparticipated in many workshops (KITPC, INT, Fermilab). He worked onthe comparison between discrete renormalization group methods thatwe have been using and continuous limits of these methods used byother authors. He was partially supported by the University as a R. A.to work on optical lattice calculations. He is working on multiflavorgauge theories and on Bs → Kµν. Now at Fermilab in AY with a URAaward. Graduation expected in June 2013.

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<P

laq

>

β

<Plaq> vs. β for Nf=4 and Nf=12

Nf=4, V=44

Nf=4, V=64

Nf=4, V=84

Nf=4, V=124

Nf=4, V=164

Nf=12, V=44

Nf=12, V=64

Nf=12, V=84

0

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1

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<>

< > vs. for Nf=4 and Nf=12

Nf=4, V=44

Nf=4, V=64

Nf=4, V=84

Nf=4, V=124

Nf=4, V=164

Nf=12, V=44

Nf=12, V=64

Nf=12, V=84


Haiyuan Zou (graduate student)

Haiyuan Zou came in fall 2008. He has passed the qualifying examand the T. A. certification. He has been working on improvedperturbation theory and complex renormalization group flows innonlinear sigma models. He has learned conventional perturbativemethods for W -production with Prof. Reno. He has been supportedpartially as a T.A. and as a R.A.

b0 = −12

(Ld − 1), b1 =18 � �� ,

b2 = − 148 "

"bb�� +

116 � �� + 1

48�� ,

b3 =1

384JJJJ�Q

Q�

�� − 1

48�� − 1

32 ""bb�� k+ 1

48 � ��

+1

32 � �� + 148��TT��

+1

24��

.


Judah Unmuth-Yockey (graduate student)

Judah Unmuth-Yockey came in fall 2011 and joined the group insummer 2012. He attended the Seattle summer school on latticegauge theory. He has been running the multicanonical code for U(1)gauge theory. He works on the Renormalization group flows in theMigdal-Kadanoff approximation and its improvement using TensorNetwork methods. He has been supported partially as a T.A. andpartially as a R.A.

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Publications in Journals

Y. Meurice, Approximate recursions for for tensor renormalization,preprint in preparation.Jon A. Bailey, A. Bazavov, C. Bernard, C.M. Bouchard, C. DeTar,Daping Du, A.X. El-Khadra, J. Foley, E.D. Freeland, E. Gamiz ,Steven Gottlieb, U.M. Heller, Jongjeong Kim (AS. , A.S. Kronfeld,J. Laiho, L. Levkova, P.B. Mackenzie, Y. Meurice, E.T. Neil, M.B.Oktay, Si-Wei Qiu, J.N. Simone, R. Sugar, D. Toussaint, R.S. Vande Water, and Ran Zhou, Refining new-physics searches inB → Dτν decay with lattice QCD, Phys. Rev. Lett. 109 , 071802(2012) (Editor’s Suggestion).http://prl.aps.org/abstract/PRL/v109/i7/e071802Y. Meurice, Remarks about Dyson’s instability in the large-N limit,e-Print: arXiv: 1203.2256 [hep-th].http://arXiv.org/pdf/1203.2256.pdf


Jon A. Bailey, A. Bazavov, C. Bernard, C.M. Bouchard, C. DeTar,Daping Du, A.X. El-Khadra, J. Foley, E.D. Freeland, E. Gamiz ,Steven Gottlieb, U.M. Heller, Jongjeong Kim (AS. , A.S. Kronfeld,J. Laiho, L. Levkova, P.B. Mackenzie, Y. Meurice, E.T. Neil, M.B.Oktay, Si-Wei Qiu, J.N. Simone, R. Sugar, D. Toussaint, R.S. Vande Water, and Ran Zhou, Bs → Ds/B → D semileptonic formfactor ratio and their application to BR(B0

s → µ+µ−), Phys. Rev. D85, 114502 (2012).http://prd.aps.org/abstract/PRD/v85/i11/e114502A. Bazavov, B. Berg, Daping Du, and Y. Meurice, Density of Statesand Fisher’s zeros in U(1) pure gauge theory, preprintarXiv:1202.2109, Phys. Rev. D 85 056010 (2012).Y. Liu and Y. Meurice, Lines of Fisher’s zeros as separatrices forcomplex renormalization group flows, Phys. Rev. D 83 096008(2011).Y. Meurice and H. Zou, Complex RG flows for 2D nonlinear O(N)sigma models, Phys. Rev. D 83 056009 (2011).


Y. Meurice, R. Perry, and S.-W. Tsai, Editors of the theme issue:New applications of the renormalization group method in physics,Phil. Trans. R. Soc. A 369 (2011).Y. Meurice, R. Perry, and S.-W. Tsai, New applications of therenormalization group method in physics, a brief introduction, Phil.Trans. R. Soc. A 369 2602 (2011).A. Denbleyker, Daping Du, Yuzhi Liu, Y. Meurice, and HaiyuanZou, Fisher’s zeros as boundary of renormalization group flows incomplex coupling spaces, Phys. Rev. Lett. 104 251601 (2010).


Recent talks

Toward dynamical gauge fields on optical lattices, Y. Meurice(Program Talk) given at “Critical behavior of lattice models inatomic and molecular, condensed matter and particle physics",KITPC, July 2012.Fisher zeros and conformality in lattice models, Yannick Meurice,with A. Bazavov, B. Berg, A. Denbleyker, Daping Du, Yuzhi Liu, Y.Meurice, D. Sinclair, J. Unmuth-Yockey, Haiyuan Zou, Lattice2012.Local gauge symmetry on optical lattices? Yuzhi Liu, YannickMeurice and Shan-Wen Tsai, Poster at Lattice 2012.Y. Meurice, “Renormalization group approach of scalar fieldtheory", U. of Rochester, April 2012.


Recent talks

Y. Meurice, “Fisher’s zeros, complex RG flows and confinement inlattice models", Miami 2011, Dec. 2011.Y. Meurice, “Fisher’s zeros, complex RG flows and confinement inLGT models" , (APS-Prairie, Nov. 2011).Yannick Meurice, “QCD calculations with optical lattices?", Lattice2011, (July 2011).Y. Meurice, “Confinement, RG flows in the complex coupling planeand Fisher’s zeros", CAQCD (Minneapolis May 2011),Y. Meurice, “Confinement and Walking Coupling Constants: ARenormalization Group Point of View", Argonne Nat. Lab. (May2011).Y. Meurice, "Fisher’s zeros as the Boundary of RG flows incomplex coupling space", UCLA, October 15, 2010.Y. Meurice, "Fisher’s zeros as the Boundary of RG flows incomplex coupling space", UC Riverside, October 18, 2010.


Y. Meurice, "Fisher’s zeros as boundary of RG flows in complexcoupling space", 5th ERG Conference, Corfu, September 14,2010.Y. Meurice, "Dynamical Gauge Fields on Optical Lattices : ALattice Gauge Theorist Point of View", (poster) KITP Conference:Frontiers of Ultracold Atoms and Molecules, Oct 11-15, 2010 .Y. Meurice, "Fisher’s zeros as boundary of RG flows in complexcoupling space", Univ. of Utrecht , August 10, 2010.Y. Meurice, "Fisher’s zeros as boundary of RG flows in complexcoupling space", Lattice 2010, Villasimius, June 18 2010.Y. Meurice, "Complex RG flows", Aspen Center for Physics, June9, 2010.Y. Meurice, "Renormalization Group in the Complex Domain",Washington University, St Louis, March 17, 2010.Y. Meurice, "Complex zeros of the beta function, confinement anddiscrete scaling", New applications of the renormalization groupmethod in nuclear, particle and condensed matter physics,Institute for Nuclear Theory Seattle. February 22 - 26, 2010.


Conference organization

New Applications of the Renormalization Group Method, INTworkshop, Feb. 22-26, 2010, with M. Birse, and S.-W.Tsai ; 35participants, including 2 U. Iowa students.Critical Behavior of Lattice Models, Aspen Workshop, May 24-June 11 2010, with G. Baym, U. Schollwoeck and S.-W. Tsai; 43participants.Critical behavior of lattice models Kavli Institute for TheoreticalPhysics in China in July 24-August 31 2012. The InternationalCoordinating Board is Lu-ming Duan (U. Michigan), YannickMeurice (U. Iowa), Shan-Wen Tsai (UC Riverside), Xiao-gangWen (MIT) and Zhenghan Wang (MicrosoftQ).Aspen Center for Physics, Summer 2013, “Lattice Gauge Theoryin the LHC Era" (coming up)


Computer facilities

2003: first 16 node cluster.2006: new cluster with 8 single CPU nodes having 3.2 GHzPentium 4 processors and Gigabyte motherboards with a build-infast ethernet card .2010: new cluster with 10 nodes with 4GB of Ram, 2.33GhzCore2 Quad processors, sata hard drives. The combined costwas $3337 or $334 per node. Built by A. Denbleyker.3 Proposals of level C at FermilabNERSC (0.5M core-hours in 2012)Helium: a large cluster at the University with 3508 computingcores across 359 nodes built in 2011 using pooled grant moneyfrom various groups at the University, and is maintained by theUniversities ITS department. We have currently able to use it, andhave requested 1M hours.


Our cluster


Recent progress

1 Toward a Fisher’s zero approach of the conformal windowO(N) nonlinear sigma modelsU(1) gauge theorySU(2) gauge theorySU(3) with fermions

2 B-physics beyond the standard modelBs → µ+µ−

B → DτνB̄s → K+µ−ν̄

3 New methods in lattice field theory focused on 3D and 4D U(1)

improved perturbative methodsRG methodsNew: Tensor Network methods


Toward a Fisher’s zero approach of the conformalwindow

Fisher’s zerosO(N) nonlinear sigma modelsU(1) gauge theorySU(2) gauge theorySU(3) with fermions


Fisher’s zeros, Finite Size Scaling and the ConformalWindow

Decomposition of the partition function (Niemeijer and van Leeuwen)

Z = Zsing.eGbounded

Zsing. = e−LD fsing.

RG transformation: the lattice spacing a increases by a scale factor b

a → baL → L/b

fsing. → bDfsing.

Zsing. → Zsing.

Important Conclusion (Itzykson et al. 83)The zeros of the partition functions are RG invariant

Fisher’s zeros: zeros of the partition function in the complex β planeYannick Meurice (U. of Iowa) Theoretical Physics (Meurice) Iowa City, October 22, 2012 33 / 87

Zsing. in terms of scaling variables

We consider discrete RG transformations

Example

b = 2, for a sigma model on D-dimensional cubic lattice: 2D fields arereplaced by one blocked field

Lattice size (in a units)L→ L/b

Scaling variables (e. g. u = β − βc + . . . , note: β ∝ 1/g2)

ui → λiui

Relevant variables: λi = b1/νi ; Irrelevant variables: λj = b−ωj

RG invariance of Zsing.

Zsing. = Q({uiL1/νi}, {ujL−ωj})


Zeros for one relevant variable

For a single relevant variable u ' β − βc , we have Zsing = Q(uL1/ν).

The complex equation Z = 0 can be written as two real equations fortwo real variables and generic solutions are isolated points.

Z = 0⇒ uL1/ν = wr with r = 1,2, . . .

This implies the approximate form for the zeros:

βr (L) ' βc + wr L−1/ν

There are many examples, where these discrete solutions followapproximate lines or lay inside cusps. In the infinite volume limit, theset of zeros may (or may not) separate the complex plane into two ormore regions.For a first order transition: ν → 1/D.


“Confining" flows: 2D O(N) models in the large-N limit

Complex extension of Tomboulis picture of confinement: the RG flowsgo directly from weak coupling to strong coupling (mass gap).

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O(N) RG flows (infinite L)

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RG F

S. P.

Figure: Infinite volume RG flows (arrows). The blending blue crosses are theβ images of two lines of points located very close above and below the [−8,0]cut of the large-N “running" β(M2) in the M2 plane. Fisher’s zeros stayoutside of the blue lines (YM, PRD 80 054020).


3D U(1): no zeros near the real axis (Denbleyker)

3D U(1) is confining. There is a gap in the spectrum and the zeros.See X-G. Wen’s book p. 265 for a discussion of confinement andduality with the XY model.

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Confidence

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Figure: Fisher’s zeros for U(1) on L3 lattices (L=4, 6 and 8 from left to right)The zeros of the real (imaginary) part are represented by the blue (red)curves and the region of confidence is below the green line (zeros near orabove this line are not reliable).


4D U(1): first or second order?

U(1) on L4: the average plaquette distribution has a double peakdistribution with equal heights at a pseudo-critical βS. For small L, thedistance between the peaks slowly decreases with the volume. (PRD85 with Bazavov, Berg and Daping Du).

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s

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44

64

84

Figure: Average plaquette distribution for U(1) at βS for L= 4, 6 and 8.


4D U(1): first or second order?

In the infinite volume limit, the width of the double peak distribution ofthe average plaquette goes to a nonzero limit (latent heat) for a firstorder phase transition and to zero as an inverse power of L for asecond order transition. Better statistics for the large volumes arenecessary to discriminate between the two scenarios.

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width 0.0172+0.230/L

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Width of double peak

width 0.168*L-0.61


4D U(1) (with Bazavov, Berg and Du)

The complex zeros appear at the intersections of ReZ=0 and ImZ=0.Results obtained by integrating a reweighted density of statescalculated with multicanonical methods (arxiv 1202.2109, PRD 85)

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ImZ=0

Figure: Zeros of the Re (+, blue) and Im (x, red) part of Z for U(1) using thedensity of states for 44 and 64 lattices. Fits favor 1st order, larger volumes areneeded.


SU(2) with βAdjoint (with A. Denbleyker and Daping Du)

Figure: The Creutz-Bhanot phase diagram (Phys. Rev. D 24, 3212-3217 ).



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!A=1.0

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Im !

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Figure: Lowest zeros for βAdjoint= 0.5, 0.6, ..., 1.5. The robustness of theseresults are discussed in Daping’s Du thesis.


SU(3) with Nf = 4 and 12 (with Yuzhi Liu and DonSinclair)

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<P

laq

>

β

<Plaq> vs. β for Nf=4 and Nf=12

Nf=4, V=44

Nf=4, V=64

Nf=4, V=84

Nf=4, V=124

Nf=4, V=164

Nf=12, V=44

Nf=12, V=64

Nf=12, V=84

Average plaquette for Nf = 4 and Nf = 12 at different volumes.Yannick Meurice (U. of Iowa) Theoretical Physics (Meurice) Iowa City, October 22, 2012 44 / 87


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Nf=4, V=84

Nf=4, V=124

Nf=4, V=164

Nf=12, V=44

Nf=12, V=64

Nf=12, V=84

< Ψ̄Ψ > for Nf = 4 and Nf = 12 at different volumes.Yannick Meurice (U. of Iowa) Theoretical Physics (Meurice) Iowa City, October 22, 2012 45 / 87


0

0.01

0.02

0.03

0.04

0.05

3.6 3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6

Imβ

Reβ

Fisher’s zeros for Nf=4 and Nf=12, m=0.02

Nf=4, V=44

Nf=4, V=84

Nf=4, V=124

Nf=12, V=44

Nf=12, V=64

Nf=12, V=84

Fisher’s zeros for Nf = 4 and Nf = 12 at different volumes.


B-physics beyond the standard model

Bs → µ+µ−

B → DτνB̄s → K+µ−ν̄


Example of Weak Matrix Element: BR(Bs → µ+µ−)

Daping Du (Fudan, U. of Iowa, Fermilab, U. of Illinois) et al. PRD 85The BR(Bs → µ+µ−) is very small: (3.6± 0.4)× 10−9. An observeddiscrepancy would open a window on possible new physics.


BR(Bs → µ+µ−)

At LHCb, the branching ratio are obtained by using comparison withother normalization channels like B+

u → J/ψK+ or B0d → K+π− in the

following way:

BR(B0s → µ+µ−) = BR(Bq → X )

fqfsεXεµµ

Nµµ

NX

fdfs

= 12.88τBd

τBs

εDsπ

εDd K

f (s)0 (m2π)

f (d)0 (m2K )

a1(Dπ)

a1(DK )

NDd K

NDsπ

f (s)0 (m2π)/f (d)0 (m2

K ) = 1.046(44)stat.(15)syst.

using MILC ensembles of gauge configurations with 2+1 flavors of seaquarks, with an improved staggered action, on (at best) 283 × 96lattices with lattice spacing a ' 0.1 fermi (so LPhys. ' 0.3hc/mπc2)


B → Dτν (Daping Du)

This work also led to a second, serendipitous paper on a hint of newphysics. The contribution of scalar form factors to the semileptonicdecay B → Dτν in the standard and 2-Higgs models provide apossible interpretation for the recent discrepancy found at BaBar. Moredetail can be found in a paper that has just been published as ahighlighted PRL.



11 ‡ 36Motivation: Semileptonic Decay B ! D! "̄

! The ratio

R(D) =Br(B ! D!""̄! )

Br(B ! D#""̄e)

" Large cancellation of experimental and theoretical systematic uncertainties." H± only enters in BR(B ! D!").

! New BaBar results 1205.5442v1



34 ‡ 36Application: B ! D!", 2HDM II?

BaBar 20122!

1!

Using Lattice form factors

0.0 0.1 0.2 0.3 0.4 0.50.0

0.2

0.4

0.6

0.8

1.0

tan"!mH#"Gev$1#

R"D#


B̄s → K +µ−ν̄ (Y. Liu URA)

The goal for the stay at Fermilab is to calculate the form factors for theB̄s → K+µ−ν̄ decay mode by using publicly available gaugeconfigurations and techniques developed by the Fermilab/MILCcollaboration. This project is intended to provide a chance to predictthe shape and normalization before the currently running LHCbexperiment. It will eventually lead to a new way to determine theCabibbo-Kobayashi-Maskawa (CKM) matrix element |Vub|. From along term point of view, precise measurement of the |Vub| is essentialto the Fermilab’s potential kaon program, such as ORKA andprospects of Project X.There have been DOE supported visits to Fermilab before thebeginning of the URA fellowship in August 2012. The plan is to visitFermilab about once a month for a period of three to four days each.The request for a seven-months on site stay supported by the URAfrom mid-August 2012 to mid-March 2013 was approved. AnotherURA proposal for March to July 2013 was approved too.


Future URA

We plan to involve one or two graduate students with URA support atFermilab during the Academic Years 2013-2014 and 2014-2015. Thereare projects in lepton-nucleon scattering relevant to Project X and thatwould also allow us to take advantage of Prof. Reno expertise.The low energy neutrino scattering or the conversion processµN → eN that can be approached with a combination of effective fieldtheory aspects and lattice simulations.There are also some interesting calculations that are needed for theproposed Project X kaon physics program such as the long-distancecontributions to the rare kaon decay K → πl+l−. Currently theStandard Model estimate relies on Chiral Perturbation Theory and haslarge uncertainties, which is why this measurement doesn’t have asmuch discovery potential as the “golden mode" K → πνν̄ for whichlong-distance contributions are GIM suppressed and the StandardModel prediction is quite precise.


New methods in lattice field theory focused toward 3Dand 4D U(1) lattice gauge theory

improved perturbative methodsRG methodsNew: Tensor Network methods


Goals (Haiyuan Zou)

Using improved perturbative methods to obtain the larger orderweak and strong coupling expansion and obtain thenonperturbative contributions. TRG is our choice.Starting from less complicated models with low dimensions. E.g.1-d and 2-d O(2) models.3d and 4d U(1) cases later.


1-d O(2) model

We calculate the weak coupling expansions (g2 ∼ 1/β) of L−link 1-dO(2) model with periodic boundary conditions.(I) The standard way is using Feynman rules:

EF ≡ ln Z [β] = const + b0 lnβ + b1/β + b2/β2 + b3/β

3 + O(1/β4),

with

b0 = −12

(Ld − 1), b1 =18 � �� ,

b2 = − 148 "

"bb�� +

116 � �� + 1

48�� ,

b3 =1

384JJJJ�Q

Q�

�� − 1

48�� − 1

32 ""bb�� k+ 1

48 � ��

+1

32 � �� + 148��TT��

+1

24��

.


1-d O(2) model

All the diagrams can be evaluated exactly. The partition function is:

Zp.b.c[β] ≡ eEF = const ·β−(L−1)/2(a0 +a1/β+a2/β2 +a3/β

3 +O(1/β4))

in which

a0 = 1, a1 =L8

(1− 1

L

)2

, a2 =L2

128

(1 +

4L− 62

3L2 +28L3 −

373L4

),

a3 =L3

3072

(1 +

18L

+87L2 −

732L3 +

1735L4 − 1782

L5 +673L6

).

(II) We have an alternative way by using the asympototic behavior atlarge β:

In(β)

I0(β)≈ exp(− n2

2β)(1 + f (n,O(

1β2 )))

in which

f (n,O(1β2 )) = − n2

4β2 +−13n2 + 2n4

48β3 +−14n2 + 5n4

32β4 + · · ·


1-d O(2) model

By increasing the order of f (n,O( 1β )), we can calculate high orders of

1/β. E.g., we get the coefficients of the system with L = 36 up to order12.

Z [β]L=36 =1

786432√

2β35/2π35/2

∞∑n=0

anβ−n

The coefficients an are listed in the table:

n an n an

0 1 6 43166266039288449055246512345193381888

1 1225288 7 2974014386617590860945

7888395046188220416

2 5521355497664 8 5537870059074860658838547

6058287395472553279488

3 3379332985143327232 9 362851322344536675792456741539

141327728361583722903896064

4 7587943455281165112971264 10 688330862660182448514514762309975

81404771536272224392644132864

5 12571105207373279142657607172096 11 2286088231017615007596833842882068655

70333722607339201875244530794496


Effect of boundary conditions in 1D O(2) (withHaiyuan Zou)

At finite volume, the nonperturbative parts of the average energy arevery different for open and periodic boundary conditions

|(E − EPT )/E | ∝ e−2β(open b.c.)∝ e−βEv (periodic b.c.)

where Ev is the energy of the periodic solution of the classicalequation of motion with winding number 1 and EPT is the averageenergy calculated as a power series in 1/β using conventionalFeynman diagrams. For D ≥ 2 such calculation would requirestochastic methods (DMC (see Svistunov and Deng’s talks), SPT, ...).


Effect of boundary conditions in 1D O(2) (withHaiyuan Zou)

-5

0

5

10

15

0 5 10 15 20

-log10[|(series-exact)/exact|]

β

-log10(e-2)β+c

errors (o.b.c)

-2

0

2

4

6

0 5 10 15 20


β

-log10(e-Ev)β+c

errors (p.b.c)

Figure: o.b.c(Left): Errors of the average energy series with order 2,4,...,20;p.b.c(L = 36)(Right): Errors of the average energy series with order 2,4,...,12.


Comparison of Hadamard series (with Haiyuan Zou)

0

5

10

15

20

25

0 5 10 15 20 25 30


β

Hadamard(H) seriesModified H series (n=10)Modified H series (n=20)

Figure: Errors of different series with order 2,4,...,20.Black:Hadamard;Blue:modified Hadamard(n = 10); Red: modified Hadamard(n = 20)


1D O(2) with L = 4,8,16,32 (with Haiyuan Zou)

Haiyuan Zou (Shandong U., U. of Iowa)The zeros are very different for open (o.b.c) and periodic boundaryconditions (p.b.c):

0

1

2

3

4

0 0.5 1 1.5 2

Im

β

Re β

zeros L=4

zeros L=8

zeros L=16

zeros L=32

zeros obc

Figure: Zeros of partition function (p.b.c) with different volumes and zeros ofpartition function (o.b.c)


SU(2) with βAdjoint , β3/2, . . . (with J. Unmuth-Yockey)

The MK approximation allows us to deal with nonlinear aspects of theRG flows. In the Migdal-Kadanoff approximation, RG flows can goaround phase boundaries (not shown).

0.1 0.2 0.3 0.4 0.5 0.6! f

1.8

2.0

2.2

2.4

2.6

! A

! 0.20 ! 0.15 ! 0.10 ! 0.05"3

1.8

2.0

2.2

2.4

2.6

" A


Two-lattice matching

Starting with a theory on a lattice with (2L)D sites and a set ofcouplings {βi}, blockspinning provides a new theory on a lattice withLDsites and new effective couplings {β′i}We have the exact identity relating a 2Mx2M Wilson loop W on theoriginal lattice to a MxM Wison loop on the coarse lattice:

< W2Mx2M >2L,{βi}=< WMxM >L,{β′i }

If you can calculate the Wilson loops numerically using MC, you canfine-tune the {β′i} on the coarse lattice in order to match the values onthe fine lattice. This is done with a finite number of couplings (oftenone) and provides an approximate discrete flow of {βi}.For spin models, the matching can be applied between correlationsof blocks of size 2M and M


2-lattice matching using Migdal-Kadanoff (with AlanDenbleyker and Judah Unmuth)

Is the MK approximation reliable? The MC calculation of 2Mx2MWilson loops for a (2L)4 lattice and the MxM Wilson loop on a LD

lattice with effective couplings obtained by the MK recursion show thatthe matching is not very accurate. LPA improvement is needed!

Volume b βF βA β3/2 β2 Psize 〈P〉 σ

84 2.40000 0.00000 2x2 0.7766 0.0067244 2 0.955274 -0.0496152 0.003759328 -0.000310275 1x1 0.7710 0.0122684 2.40000 0.00000 4x4 0.9009 0.0900744 2 0.955274 -0.0496152 0.003759328 -0.000310275 2x2 0.9973 0.0128384 4.80000 0.00000 2x2 0.4016 0.0036944 2 4.47578 -0.728286 0.188086 0.055336 1x1 0.2225 0.0065584 4.80000 0.00000 4x4 0.5670 0.1284144 2 4.47578 -0.728286 0.188086 0.055336 2x2 0.5144 0.01799


Migdal-Kadanoff Renormalization FlowsOverview

M-K RG relies upon bond moving, and re-summingFirst every other bond in one dimension is shifted over oneContinue this process for each dimensionNow sum over the “weakened” sitesThe result is a lattice with modified coupling and double the latticespacing.


Migdal-Kadanoff Renormalization FlowsThe Essentials

Migdal and separately Kadanoff have devised a coarse grainingmethod for lattice renormalization.∫

dV χr (UV )χs(W †V ) =δrs

drχr (UW ) (1)

e−Sp(U,a) =∑

r

Fr (a)drχr (U) (2)

Fr (a) =1dr

∫dU e−Sp(U,a)χ∗r (U) (3)

e−Sp(U,λa) =

[∑r

Fr (a)λ2drχr (U)

]λd−2

(4)


Migdal-Kadanoff Renormalization FlowsThe Recursion

To make the recursion formulae we insert ?? into ??.

e−Sp(V ,λa) =

[∑r

(1dr

∫dU e−Sp(U,a)χ∗r (U)

)λ2

drχr (V )

]λd−2

(5)

Using this, we can start with some inital action, S0(U,a) andpreform recursions for subsequent actions S(U, λa).Looking at the flows of β in multiple dimensions (representations),there appears to be a fixed point at zero.The flows also appear to avoid the phase transition line.After each iteration, M-K RG guarantees a supremum for thepartition function.


The Problem of Confinement for Pure GaugeYang-MillsJudah Unmuth-Yockey

An improved M-K RG scheme has been used in an attempt atproving confinement (e.g. see Tombulis (2007)).However, currently such a proof implies that U(1) is confining forall values of β.We are looking into ways to improve MK using TNRG to gaininsight in proofs of confinement.Improvments have been found by using “the two-stateapproximation”, where the model is mapped into itself, similar toM-K RG, however using tensor networks.


Complex RG flows using two lattice matching

-4

-2

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0 2 4 6 8 10

Im

Re

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0

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0 2 4 6 8 10

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2

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0 2 4 6 8 10

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-4

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0

2

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0 2 4 6 8 10

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-4

-2

0

2

4

0 2 4 6 8 10

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-4

-2

0

2

4

0 2 4 6 8 10

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Re

D=1.9,n=4 vs n=5

-4

-2

0

2

4

0 2 4 6 8 10

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Re

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Zeros n=4Zeros n=5

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-2

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Zeros n=4Zeros n=5

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RG flowsZeros n=4Zeros n=5

Figure: Complex RG flows for "log-deformed" hierarchical models. See Y. Liu,YM, H. Zou, arXiv:1112.3119, POS Lattice 2011 246.


Continuum limit of a discrete RG transformation

The Hierarchical Model recursion formula can be extended for anarbitrary scale factor b. The Fourier transform of the recursion formula

Rn+1(k) = Cn+1 e−12β

∂2

∂k2(

Rn(√

c/4 k))2

,

becomes (2 = bD)

Rn+1(k) = Cn+1 e−12β

∂2

∂k2(

Rn(b−(D+2)/2 k))bD

,

In the limit b → 1, and after some transformations, one obtains the WPequation

∂V∂t

= DV + (1− D2

)φ∂V∂φ− (

∂V∂φ

)2 +∂2V∂φ2

known to be equivalent to the optimal ERG LPA (see YM J. Phys. A 40R39-102 for Refs.)


The optimal ERGLPA

Using the basic ERG equation for a N = 1 scalar in D dimensions

∂t Γ =12

Tr∂tRk

Γ(2)k + Rk

with the LPA ansatz Γk =∫

dDx(Uk + 12(∂φ)2) we obtain

∂tu = −Du + (D − 2)ρu′ + C∫ ∞

0dyyD/2 ∂t r

y(1 + r) + u′ + 2ρu′′

where u, ρ and r are suitably rescaled versions of U, φ2 and R. UsingLitim’s optimal cutoff function r = (1/y − 1)θ(y − 1) and morerescalings, one obtains the canonical form

∂tu = −Du + (D − 2)ρu′ +1

1 + u′ + 2ρu′′


Small difference between the optimal and HMexponents

A very nice feature of the LPAs is that they allow very accuratecalculations.

The exponents for the HM and optimal LPA are very close:νHM = 0.649570365νopt . = 0.649561773(Litim; Bervillier, Juttner and Litim)

This is far from the conventional Ising universality class:νIsing3 ' 0.6304

The exponents for the HM with bD = 3, 4, 5, .... are also close and canbe calculated accurately (with Y. Liu and B. Oktay). In the following wecall this series of exponents the “discrete series".

The calculations for bD noninteger are numerically unstable for reasonsthat can be identified by considering the calculation at bD = 2 + ε.


Attempt to fit the discrete series with ERG LPAs

We consider the family of LPAs

∂tu = −3u + ρu′ +1

4π2

∫ ∞0

dy−y

52 r ′(y)

y(1 + r) + u′ + 2ρu′′

with cutoff functions r(y) = B( 1y − 1)θ(1− y) (B = 1 is optimal, see

Litim PRD 76 105001 )

Expanding in B − 1, truncating, and rescaling the first two correctionsindependently, the RHS becomes

−3u + ρu′ +1

1 + w+ ε1

1(1 + w)2 + ε2

1(1 + w)3

with w = u′ + 2ρu′′ and where ε1 and ε2 are now small independentparameters. In the following, we get a series of ν and ω by changing ε1from -0.03 to 0.02 with step 0.005 and ε2 from 0 to 0.02 with step0.0005 (Yuzhi Liu)


ν and ω for LPA(ε1, ε2) and HM bD = 2, 3, · · · 8 (Y. Liu)

0.6494 0.6496 0.6498 0.6500 0.6502 0.6504 0.6506 0.6508!0.6540

0.6545

0.6550

0.6555

0.6560"

Critical exponents ! and "

Figure: ω versus ν for the HM bD = 2, 3, · · · 8 (red) and LPA(ε1, ε2) (blue).


Decimation

In this approach, we repeatedly integrate over a fraction of the φx . Thisworks in D = 1 or by “bond sliding” approximations (Migdal).Example: 1-D Ising, 2n sites and periodic boundary conditions:Z =

∑{σi=±1} eβ

∑j σjσj+1

Using eβσ = coshβ + sinhβσ (for σ = ±1)∑{σ1=±1}

(coshβ + sinhβσ0σ1)(coshβ + sinhβσ1σ2)

= 2(cosh2 β + sinh2 β σ0σ2)

Factoring out the cosh ’s we get tanhβ′ = tanh2 β

In arbitrary dimension, it is possible to use this representation to writethe partition function as a sum over link configurations. The links cantake the values 0 and 1 with "current conservation modulo 2".


Tensor Network Formulation


Tensor Network Coarse Graining


O(2) Tensor


Numerical Results for O(2)

0 . 5 1 . 0 1 . 5 2 . 0- 2 . 0

- 1 . 5

- 1 . 0

- 0 . 5

0 . 0

T e m p e r a t u r e

E n t r o p y F r e e E n e r g y I n t e r n a l E n e r g y

O ( 2 ) m o d e l o n s q u a r e l a t t i c e ( D = 4 9 , n = 2 4 )

Figure: Free energy, entropy and average energy for the O(2) model, withTao Xiang, Zhiyuan Xie and Yuzhi Liu.


Specific Heat for O(2)

0 . 5 1 . 0 1 . 5 2 . 00 . 3

0 . 6

0 . 9

1 . 2

1 . 5

Speci

fic He

at

T e m p e r a t u r e

O ( 2 ) m o d e l o n s q u a r e l a t t i c e ( D = 4 9 , n = 2 4 )

Figure: Specific heat for the O(2) model, with Tao Xiang, Zhiyuan Xie andYuzhi Liu.


Difference with MC (Alan Denbleyker)

0.0 0.2 0.4 0.6 0.8 1.00.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

0.1 TRG and MC

MCTRG


Approximate recursions for tensor renormalization

We proposed approximate recursion formulas for Ising modelsbased on two-state truncations of the Tensor RenormalizationGroup (TRG) approach of classical lattice models. In twodimensions,we consider the cases of an isotropic blocking (as inthe Migdal recursion) and an anisotropic blocking (as in theKadanoff version) with the two state projection based on a higherorder singular value decomposition (HOSVD) used by T. Xiang etal. We also consider a projection based on a 2 by 2 transfermatrix.The transformation can be expressed as a map with 3 and 4parameters in the isotropic and anisotropic cases respectively.Linear analysis near the nontrivial fixed point yields ν = 0.987,0.964 and 0.993 for the three maps respectively, which is muchcloser to the exact value 1 than 1.338 obtained in the Migdal andKadanoff approximation.The method can be applied to other models (3D Ising and modelswith lattice fermions).


Improvement of the HM LPA: restoration of translationsymmetry?

The LPA associated with block spinning procedure requires that weisolate the block from its environment while performing the integrations.This typically create “walls" that break translational invariance but allowto do the calculation. This generates hierarchical basis for the fieldconfigurations that can be analyzed with some “tree symmetry".

I propose to try to improve the LPA by breaking the tree symmetry littleby little while restoring the translational invariance

The success of the tensor network formulation can be explained by thefact that the states are attached to the links. They are either inside theblock and summed over or piercing the boundary and kept free.


LGT calculations on Optical Lattice?

The possibility of trapping polarizable atoms or molecules in aperiodic potential created by crossed counterpropagating laserbeams has been an area of intense activity in recent years.It is now possible to physically build lattice systems where thenumber of particles and their tunneling between neighbor sites ofthe lattice can be adjusted experimentally.This opens the possibility of engineering experimental setups thatmimic lattice Hamiltonians used by theorists (e.g. theBose-Hubbard model) and to follow their real time evolution.The versatile technology of cold atoms confined in optical latticesallows the creation of a vast number of lattice geometries andinteractions.


theoretical physics part ii: lattice gauge theorymeurice/doe/ymeuricedoevisit2012.pdflattice models...

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